Minimizing Escape Probabilities: A large Deviations Approach

This paper considers the problem of controlling a possibly degenerate diffusion process so as to minimize the probability of escape over a given time interval. It is assumed that the control acts on the process through the drift coefficient, and that the noise coefficient is small. Developing a large deviations type of theory for the controlled diffusion produces several results. The limit of the normalized log of the minimum exit probability is identified as the value I of an associated (deterministic) differential game. Furthermore, we identify a deterministic (and ε-independent) mapping g from the sample values ε w(s), 0 s t, into the control space such that if we define the control used at time t by u(t) = g(ε w(s),0 s t), then the resulting control process is progressively measurable and (δ-optimal (in the sense that the limit of the normalized log of the exit probability is within δ of I).